Download Markov Regime Switching Models

Markov Regime Switching Models
• Generalisation of the simple dummy variables
approach
• Allow regimes (called states) to occur several
periods over time
• In each period t the state is denoted by st .
• There can be m possible states: st = 1,... , m
Markov Regime Switching Models
Example: Dating Business Cycles
– Let yt denote the GDP growth rate
- Simple model with m= 2 (only 2 regimes)
yt = 1 + et when st = 1
yt = 2 + et when st = 2
e t i.i.d. N(0,2)
- Notation using dummy variables:
yt = 1 D1t + 2 (1-D1t) + et
where
D1t = 1 when st = 1,
= 0 when st = 2
Markov Regime Switching Models
• How does st evolve over time?
– Model: Markov switching :
P[st  s1, s2, ..., st-1] = P[st  st-1]
– Probability of moving from state i to state j:
pij = P[st =j st-1=i]
Markov Regime Switching Models
• Example with only 2 possible states: m=2
• Switching (or conditional) probabilities:
Prob[st = 1  st-1 = 1] = p11
Prob[st = 2  st-1 = 1] = 1 - p11
Prob[st = 2  st-1 = 2] = p22
Prob[st = 1  st-1 = 2] = 1 – p22
Markov Regime Switching Models
• Unconditional probabilities:
Prob[st = 1] = (1-p22 )/(2 - p11 - p22 )
Prob[st = 2] = 1 - Prob[st = 1]
Examples:
If p11 =0.9 and p22 =0.7 then Prob[st = 1] =0.75
If p11 =0.9 and p22 =0.9 then Prob[st = 1] =0.5
Markov Regime Switching Models
Estimation:
• Maximize Log-Likelihood
• Hamilton filter - Filtered probabilities:
 1, t = Prob[st = 1 | y1, y2, …, yt ]
 2 ,t = Prob[st = 2 | y1, y2, …, yt ] = 1-  2 ,t
for t=1, …, T
Markov Regime Switching Models
GDP growth rate
4
3
2
1
0
-1
-2
-3
-4
Markov Regime Switching Models
Estimated model:
yt = 0.96 + et when st = 1
yt = -0.52 + et when st = 2
e t i.i.d. N(0,0.782)
p11 = 0.95
p22 = 0.79
Markov Regime Switching Models
 2 ,t : filtered probability of being in state 2
1.0
0.8
0.5
0.3
0.0